To determine the equation of the line that is perpendicular to the line segment joining the points $A(1,0)$ and $B(2,3)$ and divides it at $C$ in the ratio of 1:3, we can follow these steps:

Step 1: Determine the coordinates of point $C$

Given that $C$ divides the segment $AB$ in the ratio $1:3$, we can use the section formula. The coordinates of $C(x, y)$ are given by: $x = \frac{mx_2 + nx_1}{m+n}$ $y = \frac{my_2 + ny_1}{m+n}$

Here, $A(x_1, y_1) = (1, 0)$, $B(x_2, y_2) = (2, 3)$, $m = 1$, and $n = 3$.

So, $x = \frac{1 \cdot 2 + 3 \cdot 1}{1+3} = \frac{2 + 3}{4} = \frac{5}{4}$ $y = \frac{1 \cdot 3 + 3 \cdot 0}{1+3} = \frac{3 + 0}{4} = \frac{3}{4}$

Thus, the coordinates of point $C$ are $\left(\frac{5}{4}, \frac{3}{4}\right)$.

Step 2: Determine the slope of $AB$

The slope of the line segment $AB$ is given by: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{2 - 1} = 3$

Step 3: Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of $AB$ is $3$, the slope of the perpendicular line is: $m_{\perpendicular} = -\frac{1}{3}$

Step 4: Determine the equation of the perpendicular line

The perpendicular line passes through the point $C\left(\frac{5}{4}, \frac{3}{4}\right)$ and has a slope of $-\frac{1}{3}$.

Using the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$

Substitute $y_1 = \frac{3}{4}$, $x_1 = \frac{5}{4}$, and $m = -\frac{1}{3}$: $y - \frac{3}{4} = -\frac{1}{3}\left(x - \frac{5}{4}\right)$

Step 5: Simplify the equation

Multiply through by 12 to clear the fractions: $12 \left(y - \frac{3}{4}\right) = 12 \left(-\frac{1}{3} \left(x - \frac{5}{4}\right)\right)$ $12y - 9 = -4(x - \frac{5}{4})$ $12y - 9 = -4x + 5$ $12y = -4x + 14$ $4x + 12y = 14$

Thus, the equation of the line is: $4x + 12y = 14$

Would you like further details or have any questions?

Here are some questions you might consider next:

1. How do you find the equation of a line given two points?
2. What is the section formula in coordinate geometry?
3. How do you find the slope of a line perpendicular to a given line?
4. How do you convert the point-slope form to the general form of a line equation?
5. What is the importance of the slope in determining the orientation of a line?

Tip: Always remember that the slope of a perpendicular line is the negative reciprocal of the original line's slope.